1. Clustered-dot-ordered Dither Bui Hai Thanh

2. Introduction Ordered dither Fixed pattern of number (halftone cell) Two approaches Dispersed: turn on the pixel individually Clustered: group pixels to clusters Dispersed/Clustered: Depends on devices capability of displaying dots

3. Introduction Clustered ordered Described by Screen angle Screen frequency Dot pattern Level assignment

4. Introduction Screen angle Printing industry: clockwise fr. vertical line Fourfold symmetry: 0..90 O Color case: 45 O K, 75 O C, 15 O M, 0 O Y Screen frequency:# of halftone cell in a unit of length (e.g.: lpi) Dot pattern: fill-in order for dot growth sequence

5. Introduction Important development Screen functions Threshold array Holladay algorithm Rational/Irrational tangent screens Supercell Multicenter dot

6. Threshold Array Discrete representation Dot pattern in the form of threshold values 2D signal Use threshold to decide if the pixel should be turned on Goal No moiré No contour Sharp and fine details

7. Threshold Array 51181605817 4820123217099 129211252242150 8919122118168 387814010827

8. Screen Functions Guide for creating threshold array Mathematical functions are used Round dot: s(x,y)= 1-(x 2 +y 2 ) 0 1

9. Screen Functions Within each ring, pixels have to be in sequence Counter clockwise spiral Classical spiral 4561 1213147 1116158 31092 48 1 1113145 716159 31262

10. Clustered Dot Design Minimize edge/area ratio Minimize dot center migration Dot join strategy Boom dot Kiss dot Cell boundary: squared vs. round Dot growth: make the dot asymmetric Dot gain

11. Some special requirements Smooth transition: The white area on black when coverage >50% should be the negative image of black area on white s(x,y)=c x cos(2  v x x)+ c y cos(2  v y y) s(x,y)=0.5[1+sin(  v x x)sin(  v y y)] v x,v y : are the scaling factors for screen size c x, c y : aspect ratio without affecting screen frequency The negative image has a shift from positive

12. Tone level With M x N screen: MN+1 level Tone level assignment may have Uniform (linear density function) Non-uniform (non-linear density function)

13. Threshold operation The darkest element growths to the percentage of average intensity Threshold by comparing with average value. Turn on pixel with threshold level > pixel value Direct comparison

14. Holladay Algorithm Using general parallelogram Width L Height H Shift S=L-D Area: (x+u)(y+v)-xy-uv=xv+yu

15. Holladay Algorithm Halftone cell can be described by an equivalent rectangular cell Width L Height H Shift S

16. Postscript Halftone

17. Partial Dotting Mixture of full dots Provide details and preserve edges

18. Rational/Irrational Tangent Screens Screen is rational if tangent angle is a ratio of integers Have exactly the same size and shape Limited # of angle/frequency combinations Irrational tangent screen Size and shape may vary Can have more combinations

19. Rational screen Nishikawa

20. Irrational screen Can have non integer ratios So can have any angle/frequency One or more corners are not located on junctions of grid. Cost: Different in size and shape Dot pattern and # tone level vary from cell to cell Select screen with frequency, rotate on the fly If pixel in digital grid falls between points of the threshold array, interpolation is required

21. Irrational Rotation formula x’=x cos  - y sin  y’=y cos  + x sin  General expression x’=(x-x 0 ) cos  - (y-y 0 ) sin  +x 0 y’=(x-x 0 ) sin  + (y-y 0 ) cos  +y 0

22. Supercell Integer tangent has limited number Small cell provides less choice of angle Large cell causes the low frequency Divide large cell into small subcells give more choices of screen angle, with increased frequency and reduce tone level Supercell is rational cell consist of subcells, which can be varied in size/shape

23. Multicenter Dot Similar to supercell Cell is divided into equal parts Each partial dot has nucleus growing also with clustered approach within larger dot Increase frequency Drawback: slight texture contouring, and tone jump in hightlight

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